Problem: The line $ax+(a+1)y=a+2$ passes through the point $(4,-8)$. Find $a$.
Answer: Since the line passes through $(4,-8)$, we know the equation will be satisfied when we plug in $x=4$ and $y=-8$. This gives

\begin{align*}
a(4)+(a+1)(-8)&=a+2\\
4a-8a-8&=a+2\\
-4a-8&=a+2\\
-10&=5a\\
-2&=a.
\end{align*}Thus $a=\boxed{-2}$. The equation is $-2x-y=0$, or $y=-2x$, and we can see that $(4,-8)$ lies along this line.